Complex Normal Research Articles

Structural anomaly detection based on probabilistic distance measures of transmissibility function and statistical threshold selection scheme

As a mathematical representation of the output-to-output relationship, transmissibility function (TF) has been extensively applied in structural damage detection due to its robustness to influences of the input variations. As in most engineering fields, dealing with the problem of uncertainty in TF-based feature detection is an issue of fundamental importance. In this study, a new statistical, data-driven damage detection algorithm is proposed by rigorously modelling the variability of TF without postprocessing with circularly-symmetric complex Gaussian ratio distribution. The probabilistic distance of Symmetric Kullback-Leibler (SKL) divergence between TFs under baseline condition and potential damage scenarios which can measure the dissimilarity of probability distributions for the TFs under different states are computed as a damage index (DI) to detect structural anomaly. Compared against Mahalanobis distance which has the implicit assumption that the normal condition set is governed by Gaussian statistics, the probabilistic distance measure proposed in this study can deal with the deviations in TFs not following Gaussian distribution. A statistically rigorous threshold selection scheme integrating Bayesian inference strategy and Monte Carlo discordancy test is proposed to detect the the presence of damage by accommodating the uncertainties of measurements and the probabilistic model of TF. Numerical, experimental, and field test studies are conducted to validate the potential of probabilistic distance measure of TFs in anomaly detection under ambient vibration instead of forced vibration testing. Results demonstrate satisfactory performance of the proposed approach for detecting the existence and quantify the relative damage severity from a global perspective.

Capacity analysis of cooperative amplify and forward multiple‐input multiple‐output systems

AbstractThe capacity and mutual information of amplify and forward (AF) cooperative multiple‐input multiple‐output (MIMO) systems with direct link are studied and analyzed in this article. Specifically, the capacities of space modulation techniques (SMTs)‐AF and spatial multiplexing (SMX)‐AF cooperative systems are derived, and the conditions under which the capacity is accomplished are discussed and interpreted. The impact of varying system and channel parameters on the capacity is studied and discussed. As well, the impact of the presence or the absence of the direct link on the capacity performance is discussed. It is revealed that the capacity can be attained if the distribution of both direct and cooperative links follow complex Gaussian distribution. The capacity of SMTs‐AF systems is shown to be higher than that of SMX‐AF counterparts. As well, the capacity of SMTs‐AF is shown to be independent of the fading channel distribution as the channel is a source of spatial symbols in SMTs. Additionally, it is disclosed that the capacity performance is significantly enhanced in the presence of the direct link and adding more AF relays enhances the performance.

The Impact of System Distortions and Noise on HV Backscatter and Its Relevance to Above-Ground Biomass Estimation from Spaceborne P-Band SAR Data

This article analyses the effects of system distortions (crosstalk and channel imbalance), Faraday rotation and system noise on estimates of the cross-polarized backscattering coefficient, ${\boldsymbol{\sigma }}_{{{\bf hv}}}^0$ , by a spaceborne synthetic aperture radar. Modeling the unknown system errors and noise by a joint complex Gaussian distribution allows analytic first-order approximations to the mean and variance of the error in ${\boldsymbol{\sigma }}_{{{\bf hv}}}^0$ to be derived that do not depend on the SAR operating frequency. Simulation shows these approximations to be very accurate, given the statistical model and the expected magnitudes of system errors and noise for the P-band instrument to be carried by the European Space Agency BIOMASS mission. Simulation further shows that the ${\boldsymbol{\sigma }}_{{{\bf hv}}}^0$ errors are Gaussian distributed, so their exceedance probabilities can be calculated from just the analytic expressions for the mean and variance of the errors. Exceedance probabilities for above-ground biomass (AGB) can then be calculated under a power law relation between ${\boldsymbol{\sigma }}_{{{\bf hv}}}^0{{\rm{\sigma }}_{{\rm{hv}}}}$ and AGB that is consistent with P-band observations. This allows tradeoff curves between crosstalk and channel imbalance (shown to be segments of hyperbolas) to be calculated, along which the relative error in AGB is within a given percentage of its true value, from which limits on the permissible size of the errors can be determined if BIOMASS mission requirements are to be met.

RATIO-BASED SIMILARITY CRITERIA FOR POLARIMETRIC SAR IMAGE

Abstract. Dealing with multi-look polarimetric synthetic aperture radar (PolSAR) images requires averaging several independent looks to generate a sample covariance matrix of similar target scattering vectors. Along this, estimation of optimal similarity between target scattering vectors is still an open issue. In the literature, this intrinsic task has been mainly addressed in the information-based, geometric-based and detection-based frameworks. However, the derived measures mainly rely on the model assumption such as fully developed speckle and circular complex Gaussian distribution of the scattering vectors, which may not be held in high-resolution images of urban environments. To cope with this possible issue a discriminative model-free measure is proposed, where the similarity of target scattering is computed in the framework of non-local or patch based algorithm. In particular, the discriminative measure is constructed using the ratio between two pre-estimated covariance matrices of the scattering vectors. Experimental validation of the proposed measure is provided using ALOS-PALSAR image and compared with existing criterions in the literature.

The Riemann zeta function and Gaussian multiplicative chaos: Statistics on the critical line

We prove that if $\omega $ is uniformly distributed on $[0,1]$, then as $T\to \infty $, $t\mapsto \zeta (i\omega T+it+1/2)$ converges to a nontrivial random generalized function, which in turn is identified as a product of a very well-behaved random smooth function and a random generalized function known as a complex Gaussian multiplicative chaos distribution. This demonstrates a novel rigorous connection between probabilistic number theory and the theory of multiplicative chaos—the latter is known to be connected to various branches of modern probability theory and mathematical physics. We also investigate the statistical behavior of the zeta function on the mesoscopic scale. We prove that if we let $\delta _{T}$ approach zero slowly enough as $T\to \infty $, then $t\mapsto \zeta (1/2+i\delta _{T}t+i\omega T)$ is asymptotically a product of a divergent scalar quantity suggested by Selberg’s central limit theorem and a strictly Gaussian multiplicative chaos. We also prove a similar result for the characteristic polynomial of a Haar distributed random unitary matrix, where the scalar quantity is slightly different but the multiplicative chaos part is identical. This says that up to scalar multiples, the zeta function and the characteristic polynomial of a Haar distributed random unitary matrix have an identical distribution on the mesoscopic scale.

Multiband fusion inverse synthetic aperture radar imaging based on variational Bayesian inference

Images from high-resolution inverse synthetic aperture radar (ISAR) can provide more information about the targets. Multiband fusion imaging techniques can achieve higher range resolution without increasing hardware costs. A multiband fusion imaging algorithm based on variational Bayesian inference (VBI) is proposed to improve the range resolution of ISAR images. First, a multiband fusion ISAR imaging model is established based on sparse representation. Second, the scattering coefficients and noise are assumed to be the Laplacian scale mixture distribution and the complex Gaussian distribution, respectively. Finally, the fusion image is directly reconstructed in the complex domain by the VBI based on Laplace approximation method. The effectiveness and robustness of the proposed algorithm are verified by the experimental fusion results of one-dimensional signals and two-dimensional ISAR images.

Parameter Estimation of Fluctuating Two-Ray Model for Next Generation Mobile Communications

This paper focuses on the problem of parameter estimation for a Fluctuating Two-Ray (FTR) model in the context of wireless mobile communications. Precisely, the received signal is assumed to be the superposition of two dominant components (typically a direct plus a reflected path signal) in addition to diffusive secondary contributions. The signal components may be affected by random amplitude shadowing, statistically modeled by Nakagami- $m$ distribution, multiplied by unknown scaling factors with random uniform independent phases, whereas the diffusive component is assumed to follow the complex Gaussian distribution. Exploiting the method of moments, a $4\times 4$ nonlinear system is herein mathematically derived, which is very hard to be solved due to the strong nonlinearity. Therefore, a sequential procedure based on some prior information about the diffusive component power level is devised to solve it. The effectiveness of the proposed estimation technique is shown by evaluating the normalized root mean square errors as well as mean errors and standard deviations in several operating conditions of practical interest also considering the limit case of only one-ray in order to compare the proposed approach to simpler estimators, already presented in the literature. The results show the robustness of the new estimator even under a multipath model mismatch. Finally, the effectiveness of the proposed estimation procedure is confirmed through measured mmWave data.

On the Distribution Modeling of Heavy-Tailed Disk Failure Lifetime in Big Data Centers

It has become commonplace to observe frequent multiple disk failures in big data centers in which thousands of drives operate simultaneously. Disks are typically protected by replication or erasure coding to guarantee a predetermined reliability. However, in order to optimize data protection, real life disk failure trends need to be modeled appropriately. The classical approach to modeling is to estimate the probability density function of failures using nonparametric estimation techniques such as kernel density estimation (KDE). However, these techniques are suboptimal in the absence of the true underlying density function. Moreover, insufficient data may lead to overfitting. In this article, we propose to use a set of transformations to the collected failure data for almost perfect regression in the transform domain. Then, by inverse transformation, we analytically estimated the failure density through the efficient computation of moment generating functions, and hence, the density functions. Moreover, we developed a visualization platform to extract useful statistical information such as model-based mean time to failure. Our results indicate that for other heavy-tailed data, the complex Gaussian hypergeometric distribution and classical KDE approach can perform best if the overfitting problem can be avoided and the complexity burden is overtaken. On the other hand, we show that the failure distribution exhibits less complex Argus-like distribution after performing the Box-Cox transformation up to appropriate scaling and shifting operations.

Improved Shrinkage Estimator of Large-Dimensional Covariance Matrix under the Complex Gaussian Distribution

Estimating the covariance matrix of a random vector is essential and challenging in large dimension and small sample size scenarios. The purpose of this paper is to produce an outperformed large-dimensional covariance matrix estimator in the complex domain via the linear shrinkage regularization. Firstly, we develop a necessary moment property of the complex Wishart distribution. Secondly, by minimizing the mean squared error between the real covariance matrix and its shrinkage estimator, we obtain the optimal shrinkage intensity in a closed form for the spherical target matrix under the complex Gaussian distribution. Thirdly, we propose a newly available shrinkage estimator by unbiasedly estimating the unknown scalars involved in the optimal shrinkage intensity. Both the numerical simulations and an example application to array signal processing reveal that the proposed covariance matrix estimator performs well in large dimension and small sample size scenarios.

Phase Retrieval by Alternating Minimization With Random Initialization

We consider a phase retrieval problem, where the goal is to reconstruct a $n$-dimensional complex vector from its phaseless scalar products with $m$ sensing vectors, independently sampled from complex normal distributions. We show that, with a random initialization, the classical algorithm of alternating minimization succeeds with high probability as $n,m\rightarrow\infty$ when ${m}/{\log^3m}\geq Mn^{3/2}\log^{1/2}n$ for some $M>0$. This is a step toward proving the conjecture in \cite{Waldspurger2016}, which conjectures that the algorithm succeeds when $m=O(n)$. The analysis depends on an approach that enables the decoupling of the dependency between the algorithmic iterates and the sensing vectors.

Fast Multichannel Nonnegative Matrix Factorization With Directivity-Aware Jointly-Diagonalizable Spatial Covariance Matrices for Blind Source Separation

This article describes a computationally-efficient blind source separation (BSS) method based on the independence, low-rankness, and directivity of the sources. A typical approach to BSS is unsupervised learning of a probabilistic model that consists of a source model representing the time-frequency structure of source images and a spatial model representing their inter-channel covariance structure. Building upon the low-rank source model based on nonnegative matrix factorization (NMF), which has been considered to be effective for inter-frequency source alignment, multichannel NMF (MNMF) assumes source images to follow multivariate complex Gaussian distributions with unconstrained full-rank spatial covariance matrices (SCMs). An effective way of reducing the computational cost and initialization sensitivity of MNMF is to restrict the degree of freedom of SCMs. While a variant of MNMF called independent low-rank matrix analysis (ILRMA) severely restricts SCMs to rank-1 matrices under an idealized condition that only directional and less-echoic sources exist, we restrict SCMs to jointly-diagonalizable yet full-rank matrices in a frequency-wise manner, resulting in FastMNMF1. To help inter-frequency source alignment, we then propose FastMNMF2 that shares the directional feature of each source over all frequency bins. To explicitly consider the directivity or diffuseness of each source, we also propose rank-constrained FastMNMF that enables us to individually specify the ranks of SCMs. Our experiments showed the superiority of FastMNMF over MNMF and ILRMA in speech separation and the effectiveness of the rank constraint in speech enhancement.

Energy Efficient Full-Duplex UAV Relaying Networks Under Load-Carry-and-Delivery Scheme

In this paper, an energy-efficient full-duplex (FD) unmanned aerial vehicle (UAV) relaying network is proposed, where UAV acts as a mobile relay and assists information exchange between two transceivers. Specifically, the load-carry-and-delivery scheme is applied to positively take advantage of the time-varying channel gain in delay-tolerant networks; meanwhile, the FD communication policy is used to potentially further increase the energy efficiency (EE). In particular, the self-interference channel gains follow the complex Gaussian distribution instead of being constant. The EE is first rigorously derived and then, the optimum flight speed is determined under the information causality constraint to maximize the EE. Numerical results demonstrate that the proposed scheme outperforms the half-duplex as well as static schemes in terms of the EE. In addition, the impact of the self-interference cancellation factor on the EE is also demonstrated, which provides valuable insights for the system design of UAV-assisted relaying networks.

Stereo Acoustic Echo Cancellation Based on Maximum Likelihood Estimation With Inter-Channel-Correlated Echo Compensation

This paper presents batch and online algorithms of a stereo acoustic echo cancellation (SAEC) method. In SAEC, the non-uniqueness problem causes performance degradation, especially for highly coherent far-end signals. In our method, this problem can be avoided without an additional decorrelation preprocessor or multi-microphones by overestimating far-end echoes and compensating for the overestimated inter-channel-correlated echo to obtain a desired echo-canceled signal. In addition, our method is based on the maximum likelihood estimation (MLE) criterion of the echo-canceled signal under the assumption that the signal in the time-frequency domain follows a zero-mean complex Gaussian distribution with a time-varying variance. Furthermore, a variable forgetting factor based on the cross spectral density (CSD) between the echo-canceled signal and a far-end echo is presented in the online algorithm to improve the convergence of adaptive filters with a high cancellation performance when converged. Experimental results under various conditions demonstrate that the proposed method can successfully perform SAEC even in the presence of inter-channel correlation, double-talk, and abrupt echo path changes.

Two notes on “Circularly-symmetric complex normal ratio distribution for scalar transmissibility functions. Part I: Fundamentals” [Mech. Syst. Signal Process. 80 (2016) 58–77

Two notes on “Circularly-symmetric complex normal ratio distribution for scalar transmissibility functions. Part I: Fundamentals” [Mech. Syst. Signal Process. 80 (2016) 58–77

Comparison of sparse Bayesian learning and atomic norm minimization in detecting passive tonal signals

The frequency estimation of tonal signals in passive sonar systems is crucial to the identification of the marine object. In this study, we introduce the frequency estimation of tonal signals using the multiple measurement vector atomic norm minimization(MMV-ANM) technique based on compressive sensing (CS) and sparse Bayesian learning (SBL) technique based on Bayesian Inference. CS techniques have been employed in identifying the tonal frequency components. It is now well-known from the theory of CS that a high-dimensional signal can be recovered from a relatively small number of measurements as long as the desired signal is sparsely represented. MMV-ANM, an extended version of CS technique for solving the basis mismatch problem, has been proposed and shown better performance in terms of resolution and stability. On the other hand, tonal signals at passive sonar systems are also estimated with sparse Bayesian learning (SBL) using MMV. The prior source amplitudes are assumed independent zero-mean complex Gaussian distribution with the unknown variances (hyperparameters). The hyperparameters are derived by maximizing the probability of given passive data, which leads to sparse tonal signal frequencies. We compare performance of SBL and MMV-ANM in estimation of passive tonal signals with synthetic and in situ data. [*corresponding author Youngmin Choo]

A Beamforming Algorithm Based on Maximum Likelihood of a Complex Gaussian Distribution With Time-Varying Variances for Robust Speech Recognition

This letter addresses acoustic beamforming for robust speech recognition. Beamforming has been demonstrated to be one of the most effective approaches for robust recognition of distant speech using multi-microphones. In this letter, we derive a beamforming method, which we refer to as the “maximum-likelihood distortionless response (MLDR)” beamformer, based on the maximum-likelihood estimation (MLE) of a linear filter, with a distortionless constraint on the steering direction, assuming that the optimal beamformer outputs in the time-frequency domain follow a zero-mean complex Gaussian distribution with time-varying variances. By optimizing the beamformer output variances as well as the filter alternately with iterative update rules, and also by using the moving average of output powers at adjacent frames for robust estimation of an output variance, the MLDR beamformer may minimize the power of a relatively accurate noise component at the output, which resulted in better recognition performance than conventional beamformers. In addition, it can be further improved by initializing the output variances to averaged powers of a neural-network(NN)-masked input signal to estimate target speech powers, which achieved even better performance than compared beamformers exploiting trained NNs.

Spatial information theory of sensor array and its application in performance evaluation

The direction of arrival (DOA) and amplitude-phase (AP) of source signal can be estimated through array signal processing, while the process of information acquisition has not been described by information theory yet. In this study, the authors propose a novel spatial information theory framework to describe the DOA and AP estimation process. Firstly, they give a sensor array system model, where the spatial information is defined mathematically. Furthermore, in a single source scenario, the derivations of theoretical expression and an asymptotic upper bound for DOA information are presented. It also proves that the AP information accords with Shannon channel capacity when the source follows complex-Gaussian distribution. In addition, when in an uncorrelated multi-sources scenario, the AP information is the sum of that of each source for Gaussian distributed sources. Finally, they propose a definition of entropy error to evaluate the information acquisition capability of a sensor array. Numerical results verify that the entropy error is consistent with mean square error and approaches Cramer–Rao bound in high signal-to-noise ratio. It also verifies the correctness of theoretical derivation and shows that the spatial information can be employed as an evaluation index to quantitatively analyse the system performance which has significant guidance.

Azimuth Superresolution of Forward-Looking Radar Imaging Which Relies on Linearized Bregman

Forward-looking radar plays an important role in many military and civilian fields. However, the problem of low azimuth resolution has restricted its applications seriously. Although many methods have been used to achieve azimuth superresolution, the traditional methods suffer from noise amplification or limited resolution under low signal-to-noise (SNR) condition. In this paper, we proposed a Bayesian deconvolution method which relies on linearized Bregman to achieve azimuth superresolution of forward-looking radar imaging. We first used the complex Gaussian distribution and Laplace distribution to describe the distribution of noise and targets, respectively, and transformed the superresolution problem into a convex optimization problem by maximum a posteriori estimation in the Bayesian framework. Second, linearized Bregman algorithm was used to solve the convex optimization problem. The proposed method introduces the prior information of noise and target, and overcomes the ill-posedness of deconvolution. As a result, the azimuth resolution is remarkably enhanced. Besides, the proposed method has high computational efficiency by linearizing objection function, so it can take both time cost and resolution improvement into consideration. Finally, the superior performance was verified by simulation and experimental data.

Edge Exclusion Tests for Improper Complex Gaussian Graphical Model Selection

We consider the problem of inferring the conditional independence graph (CIG) of improper complex-valued Gaussian vectors. A $p$ -variate improper complex Gaussian graphical model associated with an undirected graph with $p$ vertices is defined as the family of improper complex Gaussian distributions that obey the conditional independence restrictions implied by the edge set of the graph. For real random vectors, considerable body of paper exists where one first tests for exclusion of each edge from the saturated model, and then infers the CIG. Prior work on proper complex Gaussian graphical models is sparse, while that on improper complex Gaussian graphical models is non-existent. In this paper, we propose and analyze a generalized likelihood ratio test (GLRT) based edge exclusion test statistic for improper complex Gaussian graphical models. The null distribution of the test statistic is specified explicitly to allow analytical calculation of the test threshold. An alternative computationally fast version of the GLRT statistic is also derived. Simulation examples are presented to illustrate the proposed statistic.

Statistical modeling for fast Fourier transform coefficients of operational vibration measurements with non-Gaussianity using complex-valued t distribution

Abstract Frequency-domain operational vibration responses have been widely applied in structural health monitoring (SHM). Characterizing and quantifying their uncertainties are of fundamental importance for enhancing the robustness of SHM technologies. The classic complex Gaussian distribution is being increasingly used to model the distributions of fast Fourier transform (FFT) coefficients due to its elegant and convenient mathematical nature. However, the field-test data analysis of engineering structures under operational vibrations in this study emphasize the possibility of non-Gaussianity of some observations. The higher peaks and heavier tails than those of a Gaussian distribution emerge as observable features. As a member of the general family of elliptically symmetric distributions, the t distribution has been widely recognized as a useful extension of the Gaussian distribution for the robust statistical modeling of data sets with heavier-than-normal tails. In this paper, we consider using the complex-valued t distribution to characterize the FFT coefficients with high kurtosis and heavy tails. The probability density function (PDF) of multivariate proper complex t random variables are derived based on the equivalent counterparts in the real-valued domain. The marginal PDFs of the real part, the imaginary part, the magnitude, and the phase of a univariate complex-valued t random variable are also derived analytically based on advanced integration techniques. The field test data of different engineering structures provide an illustration of the performance of complex Gaussian and complex t probabilistic models at different frequencies evaluated via the K-S test, goodness-of-fit test, and probability plots.